Variations I have heard (and sing):

Barnum (&) Bailey's Circus come to town put up a tent that was big and brown

Re; The Furniture Man; He'll take anything from a Nursery plant, from a skillet to a fryin'pan

Mystery re Baby's in a cradle:

I take cradle to mean "crib". Cribs were rows of shacks in the Storyville district of New Orleans where the very cheap whores plied their trade.

Baby's in a cradle down in New Orleans, Doctor kept whiffin till my baby got mean

He kept whiffin till his nose got sore etc etc